Chapter 9: Q. 24 (page 772)

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.
$directrix x = - 1, focus (2, 5)$

Short Answer

Expert verified

The equation is $(y - 5)^{2} = 6 x - 3 .$

Step by step solution

Step 1. Given information.

It is given,

$directrix x = - 1, focus (2, 5)$

Step 2. Distance formula.

Now,

$Let (x, y) be any point on the parabola. Formula for the distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

The distance between the point and the focus is,

$Distance = \sqrt{(2 - x)^{2} + (5 - y)^{2}} [since x_{1} = x, y_{1} = y, x_{2} = 2, y_{2} = 5]$

Now the distance between the point and directrix is $| x + 1 | [since x = - 1]$

Therefore,

$\sqrt{(2 - x)^{2} + (5 - y)^{2}} = | x + 1 |$

Step 3. Final answer.

On simplifying the equation,

${(\sqrt{(2 - x)^{2} + (5 - y)^{2}})}^{2} = (| x + 1 |)^{2} (2 - x)^{2} + (5 - y)^{2} = (x + 1)^{2} (4 - 4 x + x^{2}) + (25 - 10 y + y^{2}) = (x^{2} + 2 x + 1) y^{2} - 10 y + 29 - 4 x = 2 x + 1 y^{2} - 10 y + 29 - 4 x + 4 x = 6 x + 1 y^{2} - 10 y + 29 = 6 x + 1 (y - 5)^{2} + 4 = 6 x + 1 (y - 5)^{2} + 4 - 4 = 6 x + 1 - 4 (y - 5)^{2} = 6 x - 3$

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Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.
$directrix x = - 1, focus (2, 5)$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Distance formula.

Step 3. Final answer.

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Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.directrixx=-1,focus(2,5)

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Distance formula.

Step 3. Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Decision Maths

Theoretical and Mathematical Physics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.
$directrix x = - 1, focus (2, 5)$